# Is there a monster behind the trees?

‎First Fix the following notation:‎ ‎

$‎‎\forall ‎\kappa‎\in Card~~~Tp(‎\kappa‎):="‎\kappa‎~has~tree~property"$‎ ‎‎‎‎

The large cardinals as "monsters of heaven" live everywhere in the land of mathematics. Most of them appear in topology, measure theory, infinitary logic, category, model theory, etc. But some of them live in a strange misty place, "the forest of tall trees"! One of the most important problems of set theory is to discover that which one of these monsters have a forestial nature? The forestial nature of a monster gives us very important information about his possible behavior at many combinatorial situations in set theory and model theory. Now define the following concepts:‎ ‎

Definition (1): ‎Let ‎‎$‎A‎$‎, ‎$‎‎B‎$ be ‎large ‎cardinal ‎axioms, $I‎$‎, ‎$‎J‎$‎ ‎be ‎sub ‎classes ‎of ‎‎$‎Ord‎$ , ‎‎$‎‎\lbrace ‎‎\alpha‎‎_{i}‎‎\rbrace‎_{i\in I}, ‎\lbrace ‎‎‎‎‎‎\beta‎‎_{j}‎‎\rbrace‎_{j\in J}\subseteq Ord$ , $\lbrace ‎‎‎‎‎‎\alpha‎‎_{i}‎‎\rbrace‎_{i\in I}\cap ‎‎\lbrace ‎‎‎‎‎‎\beta‎‎_{j}‎‎\rbrace‎_{j\in J}=‎\emptyset‎‎‎$

‎then respectively every ‎statement ‎in ‎the ‎form ‎‎$(‎‎‎\star)‎$ and ‎$‎‎(‎\star‎‎\star‎)$‎ ‎called a‎ ‎"weak" and "strong" tree property "equation".‎ ‎ ‎‎ $(‎\star‎)~~Con(ZFC+A+\bigwedge_{i\in I}Tp(\aleph_{‎\alpha‎_{i}}))\Longleftrightarrow‎ Con(ZFC+B+\bigwedge_{j\in J}Tp(\aleph_{‎\beta‎_{j}}))$ ‎‎ ‎

$(‎\star\star‎)~~(A+\bigwedge_{i\in I}Tp(\aleph_{‎\alpha‎_{i}}))\Longleftrightarrow ‎(‎B+\bigwedge_{j\in J}Tp(\aleph_{‎\beta‎_{j}}))$ ‎ ‎

Example (1): Note to the following ‎well ‎known ‎results:‎ ‎

‎ A‎ ‎wea‎k tree property equation: ‎ ‎‎ ‎ $Con(ZFC+Tp(\aleph_{2}))\Longleftrightarrow ‎Con(ZFC+‎\exists~a~weakly~compact~cardinal‎‎‎)‎‎‎$‎‎ ‎

‎‎ A strong tree property equation:
‎ ‎ $(‎\kappa‎‎~is~strongly~inaccessible~+~Tp(‎\kappa‎))‎\Longleftrightarrow (‎‎‎‎\kappa‎‎~is~weakly~compact)‎‎‎$‎‎

‎‎‎ ‎ Remark (1): ‎It ‎seems ‎there are ‎many weak ‎tree ‎property "‎unequalities" like the following results, ‎but strangely the weak and strong "equalities" are too rare. These equalities uncover some fundamental relations between large cardinal axioms and combinatorial properties of other cardinals and so could be very useful and valuable.

‎‎‎ A‎ ‎weak ‎tree ‎property ‎unequality ‎discoverd by Menachem ‎Magidor:‎ ‎ ‎‎ $‎Con(ZFC+ ‎‎‎\exists‎~\mathtt{0}^{\sharp}‎)‎\Longleftarrow‎ ‎‎‎Con(ZFC+Tp(\aleph_{2})+Tp(\aleph_{3}))‎‎$‎ ‎

‎‎ ‎Another ‎weak ‎tree ‎property ‎unequality ‎discoverd ‎by ‎Uri ‎Abraham:‎ ‎ ‎‎‎ ‎$‎‎‎Con(ZFC+‎\exists‎~a~supercompact~cardinal~with~a~weakly~compact~cardinal~above~it‎) ‎‎\Longrightarrow‎ ‎Con(ZFC+Tp(\aleph_{2})+Tp(\aleph_{3}))‎$‎ ‎

‎ ‎ Question (1): ‎Is ‎there ‎any ‎known large ‎cardinal ‎axiom stronger than "$‎‎‎\mathtt{0}^{\sharp}‎$ exists" ‎and ‎weaker ‎than "there is a supercompact cardinal with a weakly compact cardinal above it " ‎‎like ‎$‎‎A‎$ ‎in ‎which ‎we have an ‎"weak tree property equality" ‎in ‎Magidor ‎and ‎Abraham's ‎results? ‎What ‎about ‎"strong tree property equality"? ‎‎In ‎the ‎other ‎words‎: is there any large cardinal axioms ‎$‎A‎$, ‎$B$, ‎‎$C$‎ ‎which ‎the ‎foll‎owing statements be true?‎ ‎

‎ $(1)~Con(ZFC+A)\Longleftrightarrow ‎Con(ZFC+Tp(\aleph_{2})+Tp(\aleph_{3}))‎‎‎‎$‎ ‎ ‎‎ ‎$‎‎(2)~B \Longleftrightarrow (C+Tp(\aleph_{2})+Tp(\aleph_{3}))$‎ ‎ ‎‎

Question (2): Are there any other known weak or strong tree property equality different from statements in example ‎$‎‎(1)$?‎

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Please avoid manually adding spaces in equations, as a rule, they don't display well on devices other than your own. Also, what was the point of redefining all logical notations? The usage of + is standard for axiomatic systems but your definitions are incorrect: ZFC is not a sentence and your $\mathcal{L}$ is undefined. –  François G. Dorais Aug 27 at 13:25
Dear Francois, your critic remarks are true. About changing the logical symbols I want to add an algebraic form to the statements just based on a personal intuition from model theory. But based on your comment I think this is rather unclear and I will change them. –  Ali Sadegh Daghighi Aug 27 at 13:43

1. An interesting result of Magidor-shelah says that if $\kappa$ is a singular limit of strongly compact cardinals, then $TP(\kappa^+)$ holds.

2. For more about the necessity of the use of large cardinals for getting tree property at successive cardinals see "Foreman, Matthew; Magidor, Menachem; Schindler, Ralf-Dieter The consistency strength of successive cardinals with the tree property. J. Symbolic Logic 66 (2001), no. 4, 1837–1847."

3. A recent result of Itay Neeman says that assuming the existence of $\omega-$many supercompact cardinals, it is consistent to have tree property up to $\aleph_{\omega+1}.$

4. Gitik's paper "Extender based forcings, fresh sets and Aronszajn trees" gives the exact consistency strength of $\aleph_\omega$ being strongly limit, and $\aleph_{\omega+2}$ have tree property.

5. A recent paper of Sy Friedman and Radek Honzik "More on tree property" gives the consistency of tree property at all $\aleph_{2n}, 1\leq n< \omega$ and $\aleph_{\omega+2}.$

6. For more on tree property, in particular at successors, or double successors of singular cardinals, see the works of Itay Nemman, Dima Sinapova and Spencer Unger.

7. A recent result of Sy Friedman and Fontanella shows that we can get tree property at $\aleph_{\omega+2},$ just starting from a weakly compact cardinal. In their proof, $\aleph_\omega$ is not strong limit of course. They also have a proof of tree property at successor and double succesor of a singular cardinal $\kappa$, but again in their model $\kappa$ is not strong limit.

8. By a result of Spencer Unger, it is consistent that there are no special Arinszajn trees at all $\aleph_n, 1<n<\omega,$ assuming the existence of infinitely many Mahlo cardinals. More consistency results are proved in Unger's papepr "Fragility and indestructibility II".

9. The new paper "Combinatorial properties of successors and double successors of singular cardinals " by Fontanella-Friedman also contains some interesting results.

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Thanks for your useful references. –  Ali Sadegh Daghighi Aug 27 at 14:06
$\kappa$ is strongly inaccessible + $sTp(\kappa) \Longleftrightarrow \kappa$ is Mahlo.
Where $sTp(\kappa)$ would mean "there are no special $\kappa$-Aronszajn trees".